ASSOCIATIVITY AND COMMUTATIVITY OF PARTIALLY ORDERED RINGS

There are some classical results about automatic associativity and commutativity of ordered rings. In particular, they are established for some lattice-ordered rings in the uniformly bounded case, i.e. for f-rings and for totally ordered skew fields. These results admit a common generalization to certain ordered rings. Here the important additional assumption is "the localizability of the order". In this talk, I will present both classical and new results, and discuss a (possibly new) application to partially ordered skew fields. The key technique used in the proof is based on results about the multiplicativity of pure states. The talk is based on the preprint arXiv:2303.03627.